Eigenvalues of singular measures and Connes noncommutative integration

Abstract

For a singular measure $\mu$, Ahlfors regular of order $\alpha>0,$ with compact support in $\mathbb{R}^{\mathbf{N}}$ and a pseudodifferential operator $\mathbf{A}$ of order $-l=-\mathbf{N}/2$ we consider the compact operator $\mathbf{T}(P,\mathbf{A}) = \mathbf{A}^*P\mathbf{A}.$ Here $P$ is the signed measure, $P=V\mu$ with density $V$ belonging to the Orlicz class $L^{\Psi,\mu}$ with $\Psi(t)=(t+1)\log(t+1)-t.$ Using eigenvalue estimates for such operators, obtained in \texttt{arXiv:2011.14877}, we establish eigenvalue asymptotics of $\mathbf{T}(P,\mathbf{A})$ for a class of measures, including the ones supported on uniformly rectifiable sets. These results lead to the measurability in the sense of A.Connes of operators $\mathbf{T}(P,\mathbf{A})$ and a formula for the singular trace of these operators, producing a noncommutative version of integral with respect to singular measure.